[seqfan] Re: Is 0001 a 4-digit base 10 number ?

[israel at math.ubc.ca]

Indeed, for the base-b version, let M be the b x b tridiagonal matrix with 
1's on the main diagonal and the diagonals above and below it, 0's 
everywhere else, and e the b-dimensional column vector of all 1's, then 
a(n) = e^T M^(n-1) e for n >= 1, and the generating function is g(z) = 1 + 
z e^T (I - z M)^{-1} e, which is a rational function with poles at the 
reciprocals of the eigenvalues of M. A linear recurrence for n >= 1 is 
sum_{j=0}^b c_j a(n+j) = 0 where the characteristic polynomial of M is 
sum_{j=0}^b c_j t^j. In particular for b=10 the recurrence is
 -a[n]+4*a[n+1]+18*a[n+2]-36*a[n+3]-35*a[n+4]+84*a[n+5]-14*a[n+6]-48*a[n+7]
+36*a[n+8]-10*a[n+9]+a[n+10] = 0

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
 


On Aug 19 2011, franktaw at netscape.net wrote:

>Incidently, A126364 has a rational generating function (and 
>equivalently a linear recurrence).
>
>Note that the base 3 equivalent is A078057 (essentially the same as 
>A001333).
>
>Franklin T. Adams-Watters
>
>
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